SECT. II. STABILITY OF SOLAR SYSTEM. 11 



whence the celebrated problem of the three bodies, originally 

 applied to the moon, the earth, and the sun namely, the masses 

 being given of three bodies projected from three given points, 

 with velocities given both in quantity and direction ; and sup- 

 posing the bodies to gravitate to one another with forces that are 

 directly as their masses, and inversely as the squares of the dis- 

 tances, to find the lines described by these bodies, and their 

 positions at any given instant ; or, in other words, to determine 

 the path of a celestial body when attracted by a second body, 

 and disturbed in its motion round the second body by a third 

 a problem equally applicable to planets, satellites, and comets. 



By this problem the motions of translation of the celestial 

 bodies are determined. It is an extremely difficult one, and 

 would be infinitely more so if the disturbing action were not very 

 small when compared with the central force ; that is, if the 

 action of the planets on one another were not very small when 

 compared with that of the sun. As the disturbing influence of 

 each body may be found separately, it is assumed that the action 

 of the whole system, in disturbing any one planet, is equal to 

 the sum of all the particular disturbances it experiences, on the 

 general mechanical principle, that the sum of any number of 

 small oscillations is nearly equal to their simultaneous and joint 

 effect. 



On account of the reciprocal action of matter, the stability of 

 the system depends upon the intensity of the primitive mo- 

 mentum (N. 59) of the planets, and the ratio of their masses to 

 that of the sun ; for the nature of the conic sections in which the 

 celestial bodies move depends upon the velocity with which they 

 were first propelled in space. Had that velocity been such as to 

 make the planets move in orbits of unstable equilibrium (N. 60), 

 their mutual attractions might have changed them into parabolas, 

 or even hyperbolas (N. 22); so that the earth and planets might, 

 ages ago, have been sweeping far from our sun through the 

 abyss of space. But as the orbits differ very little from circles, 

 the momentum of the planets, when projected, must have been 

 exactly sufficient to ensure the permanency and stability of the 

 system. Besides, the mass of the sun is vastly greater than that of 

 any planet ; and as their inequalities bear the same ratio to their 

 elliptical motions that their masses do to that of the sun, their 



